= E := {A,,6,(S)+ (1- X,J)6J(S) IA,Je [0,T] and 6,(S),6J(S)EZ V}

Stability Margins for Multilinear Interval Systems via Phase Conditions: A Unified Approach

L.H. Keel Centerof Excellencein InformationSystems Departmentof ElectricalEngineering TennesseeStateUniversity Texas A&M University Nashville,TN 37103-3401 CollegeStation,TX 77843

Abstract tested.These conditionswhich are sufficientcondi- tionscan be tightenedby increasingthe number of verticesand thiseventuallyleadsto necessaryand This papergivesa simpleway ofcheckingthestabil- sufficientconditions.Using theseresultswe show itywith respectto an arbitrarystabilityregionofa how variousworstcasestabilitymarginssuchas gain, familyofpolynomialscontaininga vectorofparame- phase,time-delay,]/ooand sectorbounded nonlinear tersvaryingwithinprescribedintervals.Itisassumed stabilitymarginsformultilinearintervalsystemscan thattheparametersappearaffinemultilinearlyinthe be found. characteristicpolynomialcoefficients.The condition proposedhereissimplyto checkthephasedifference 2. NOTATION AND MAIN RESULTS of the vertexpolynomials.This testbased on the mapping theoremsignificantlyreducescomputational Let p = [Pl,/_,"",/_]denotea vectorofrealparam- complexity.We omit mathematicalproofsin this eters.Considerthepolynomial version.The resultscan be used to determinevar- iousstabilitymargins of controlsystemscontaining 6(s, p) :: 6o(p)+61(p)s+62(p)s2+ "" .+6,,(p)s". (1) interconnectedintervalsubsystems. These include wherein the coefficients 61(p) are o_ne multilinear the gain, phase, time-delay, H_ and nonlinear sec- functions of p, i = 0, 1,-..n. The vector p lies in tor bounded stability margins of multilinear interval an uncertainty set systems. II:={plp_-_

The problemofdeterminingwhethera controlsystem A := {6(s,p) IP E If}. (3) remainsstableor not when some of itsparameters LetV denotetheverticesof If,i.e., undergo perturbationswithinprescribedintervalsis one ofthecentralproblemsinrobustparametricsta- V := {p I = or Pi = pi-, i = 1, 2,..., 1) (4) bility.The casewhere theparametersappear affine linearlyin the coefficientsof thecharacteristicpoly- and nomialhas been solvedeffectivelyby the Edge The- A V :: {6(s,p) [p E V}. (5) orem [I] and the Box Theorem [2,3] The case where denotesthe setof vertexpolynomials. the coefficients are affine multilinear functions of the parameters is of current interest (see [4] - [9]). This Fixings = s*,we letA(s*) denotetheactof points case arises in control systems where transfer function 6(s*,p) in the complex planeobtainedby lettingp coefficients perturb or state space parameters perturb rangeoverH: or matrix fraction factors perturb. The present paper A(s*) := {6(s*,p)IP EII}. (6) deals with the multilinear case also and shows that a sufficient condition for the stability of such systems Likewise is that the phase difference between various vertex AV(S* ) :-- {6(s*,p) I P E V}. (7) polynomials evaluated along the stability boundary be less than 180". This is an "image set" approach The convex hull of a set of points P in the complex using the convex hull property of multilinear image planeisco 7). Denote thesetofconvexcombinations set [10,11,12]. We show how this result can be used ofthevertexpolynomiMsby in conjunction with the multilinear version of the Box and Theorem [5] to develop a highly efficient phase test E := {A,,6,(s)+ (1- x,j)6j(s) IA,jE [0,t] forstabilitywhere only the Kharitonovverticesare 6,(s),6j(s)E Z v}. (8)

(NASA-CR-192893) STABILITY MARGINS N93-25226 FOR MULTILINEAR INTERVAL SYSTEMS BY WAY OF PHASE CONDITIONS: A UNIFIED APPROACH (Tennessee State Univ.) Unclas 5 p Let 8 C (r denote an open stabilityregion and 08 the boundary of S. Let A(A) denote the image set evaluatedat s = A of the set of polynomials A. We "can now state a firstresultbased on the Mapping m Theorem of [13].

Theorem 1. The set of polynomials A i$ stable with respect to 8 if it has at least one stable poly- Figure 1. Unity Feedback System nomial and i) 0 _ A(A) for some A E Oa ii) the set of polynomials E is stable _ith respect to 3. 3.1. Gain Margin The problem of checking the stabilityofthe set E can be solved computationally relativelyeasilyin view if Consider the family offeedback systems as above with the factthat E()0 consistsof straightlineswhose end points are contained in the set AV(A ). To formalize M(s,p,q) N(s,p) (11) := D(s, q) thislet6(a)be a polynomial and s --A a point in the complex plane. Let Ca (A) denote the argument of the where At(s, p) and Dis, q) are multilinear functions complex number 6(_): of p and q and (p,q) varies in a polytope P with vertex set P¢. The upper gain margin of the family := (9) of the family of feedback systems with M is defined as the largest value L* >_ 0 of L so that For a given set of polynomials T let Dis, q) + (1 + L)N(s, p) (12) CI'T(A) := sup - ¢,,(_) I. (10) 6",,6"2ET remains FIurwitz for all/; • [0, L*) and for all (p, q) • P. The lower gain margin of the family may be sim- We can now statethe followingusefulcorollary. ilarly defined by replacing (1 + L) by (1 - £). Let us first define the following sets corresponding to each Corollary 1. (Theorem 1) The set of polynomials fixed real value of L: A is stable with respect to 8 if it contains one stable 2"L(w) :'- {D(jw, q) + (I + L)N(jw, p)l polynomial: (p,q) • 73}. (13) a) o ¢

:= co {D(jw,q) + (1+ L)N(jw, p)i b) supxeoS_AV(_ ) < x where 08 denotes the (p, q) • 73v}. (14) boundary of the region 8. Now let L(w) := inf(L J 0 C Z_(w)} (15) The above resultstates that A is stable ifthe net phase differenceof the vertex polynomials evaluated and similarly, along the stabilityboundary is lessthan 180°. We shall refer to this as the Phase Condition. The L(w) := inf{L I 0 • _(_)}. (16) Phase Condition can be easilyverifiedby the Seg- The corresponding gain margins are: ment Lemma [13]without sweeping over frequency.

L* := infL(,#) (17)

3. CALCULATION OF STABILITY MAR- £" := infi,(_) (18) GINS

Theorem 2. The gain margin L* Exact calculation of stability margins for a multilin- 4 M(s) is bounded from below by L*. ear interval control system is computationally diffi- cult. In this section, we give simple techniques to Ulin- earize" the problem so that the Phase Condition can The point here is that lower bound L* can of course be used to estimate various stability margins without be found from the Phase condition, since it involves excessive calculations. only the vertices. 3.2. Phase Margin and define

:= SimilaHy, the phase margin of the family with M(s) T(w) inf{T I 0 6IT(W)} (28) "is defined as the largest value 0* _> 0 of 0 so that _'(w) := inf{T [ 0 6_T(W)}. (29)

D(s, q) + eJ°N(s, p) (19) The corresponding maximum time delays are:

is Hurwitz for all 8 6 [0, 0*) and for all (p, q) 6 7_. T* := infT(w) (30) Consequently, we define the sets corresponding to each fixed value of 8: := inf _(_) (31) I_(w) := {D(jw,q)+ei°N(jw,p)l Theorem 4. The time delay margin T* of M(s) is (p,q)6 (20) jo} bounded from below by _'*. I÷(w) := co {D(jw,q)+d°N(iv,p)] (p,q) 6 Pv}. (21) As before _'* can be found from the Phase Condition.

And let

¢(w) := inf{¢ I 0 ez_(_)} (22) 3.4. H _ Stability Margin I ¢(w) := inf{¢ 0 e_(_)}. (23) Consider the following configuration with Q(s) fixed The corresponding phase margins are: M(s,p,q) as before and AM representing an unstructured perturbation. Let 4" := inf¢(w) (24)

¢" :- inf¢(w) (25)

Theorem 3. The phase margin ¢* of M(s) is bounded from below by ¢*.

Again this lower bound can be found using the Phase Condition.

3.3. Time-Delay Margin Figure 3. Additive Perturbations

Q(s) _ (32) = Q2Cs)

then the H _ stability margin of the system is defined as follows:

Figure 2. Time-Delay System 1 N(_,P) _ 1 suPcp, q)e_, [ Q(s)[l + Q(s) D0,q)] oo

Consider the system with the time delay T. The time 1 delay margin of the system is the maximum allow- II Q,(.)-(.,_ I " sup(P'q)e_ II Q'(s)[Do'q)+Q_(s)N("P)] ] able time delay preserving closed loop stability. It can be calculated as before from the convex hull of From the Lemma in [14], we have the corresponding image sets. Let us define the sets 1 corresponding to each fixed real value of T: Vfl> 0, IIQ(s)[l + M(s,p,q)Q(s)]-_ll_,, < -_ ZT(W) := {D(iw,q) 4-e-J'_TN(jw,p) ] for all M(s, p, q) iff (p, q) 6 P} (26)

_T(0;) :---_ {D(jw,q) +e-JWTN(j_,p) ] Q2(s)DCs, q) + Qt (s)N(s, p) + flvi°Q, (s)D(s, q) (p,q) 6 7:_,} (27) = [Q2Cs) + _ei°Q_(s)]D(s, q) + QtCs)N(s, p) isHurwitz for all0 6 [0,2_r),and (p,q) E P. Now I)Re{9(0)}> 0 letus definethe setscorresponding to each fixedreal value of a and 8: 2) n(s) isIIurwitzstable {[Q2(jw)+ 1_eJSQ1(jw)]D(jw,q)T 3) d(s)+ jan(s) is Iiurwitsstablefor alla 6 R. Ql(jW)N(jw,p)I(P,q)6 _} (33) Consider co{[Q2(jw)+ BeiaQ1(jw)]D(jw,q)+ Qx(jw)N(jw, p) [ (p,q) 6 T)v}. (34) 1 1 N(s, p) -_ + M(jw, p, q) : _ + D(:,q) If we let D(s,q) + klV(s,p) (40) = kD(s,q) " _(_) .= inf{fl I 0 6 27a,0(w) for some 0} (35) $(_) := inf(fl I 0 6:2a,0(w) for some 0} (36) Suppose the transfer function satisfies the above two conditions: The corresponding stabilitymargins are"

1) Re Sv(°'q)+kN(°'P)'[ /Y = _ff_(w) and /_*= inf,(w). (37) kD(O,q) J > 0 for all k • [0, k*] and for (p, q) • "P

Theorem 5. The H _ stability margin _* of the 2) D(s, q) + kN(s, p) • 7g forallk • [0,k'] and for configuration in Figure 3 is bounded from below by (p,q) • 9. _*.

Let us define the following setscorresponding to each The bound/_* can be found from the Phase Condi- fixed value of k and a: tion. _'a,,,(w) := {kD(jw,q)+j_[D(jw,q) +kN(jw, p)] I(P, q)• 7)} 3.5. Nonlinear Sector Bounded Margin = {k[D(jw,q)+ ]o_N(jw,p)] +jaD(flo, q) [(p, q) • "P} (41) Ik,,,(o;) := co {k[D(j_,q)+ jaN(jw,p)] u(,) +j_D(jw, q) I (P,q) • Pv}(42)

If we let

k(w) := inf{k I o 62"k,,,Cw),for some a} (43) k(w) := inf{k ] 0 62"t,,,(w),for some a},(44) Figure 4. LqardProblem the corresponding nonlinear sector bounded margins are: Now letus considerLurd problem. According to [15], ifg(s) isa stabletransferfunction and _ belongs to k* = inf k(w) and ;2 = inf k(w). (45) the sector [0,k],then a sufficientconditionfor abso- lutestability is Theorem 6. The nonlinear sector bounded margin k* of M(s) is bounded from below by k*. Re{_+g(jw)}>0, V w6R. (38) 4. CONCLUDING REMARKS In this problem we want to find the maximum value k* of k so that We have given a simple computational method to check the stability and compute various stability mar- Re{l+M(j_,p,q)}>O, V _aER (39) gins of polynomial families containing parameters which appear afflne multilinearly in the character- for all M(s,p,q). istic polynomial coefficients and which vary in pre- scribed intervals. Furthermore, the proposed method From the theorem given in [15], we know that g(s) = completely eliminates frequency sweeping. Although _-_is SPR iff the following three conditions are sat- the testisonly a sufficientconditionwe expect it to isfied: be usefulespeciallyin view of the practicalnecessity of maintaining adequate margins of stability. Addi- [9] B. D. O. Anderson, F. Kraus, M. Mansour, and tionedly the bounds obtained on the stability margins S. Dasgupta, "Easily testable sufficient condi- can be made as exact as desired by introducing addi- tions for the robust stability of systems with mul- ¢ional vertices. Finally it is worth emphasizing that tilinear parameter dependence,". Submitted for the computations given here avoid construction of im- publication. age sets,a task which isenormously unwieldy from a [10] R. R. E. De Gaston and M. G. Safonov, "Ex- computational standpoint. act calculation of the multiloop stability margin," IEEE Transactions on Automatic Control, vol. AC - 33, pp. 156 - 171, February 1988. References [11] S. Dasgupta, P. J. Parker, B. D. O. Anderson, [1] A. C. Bartlett, C. V. Hollot, and H. Lin, "Root F. J. Kraus, and M. Mansour, "Frequency do- location of an entire polytope of polynomials: it main conditions for the robust stability of linear suffices to check the edges," Mathematics of Con- and nonlinear dynamical systems," IEEE Trans- trola, Signal8 and Systems, vol. 1, pp. 61 - 71, actions on Circuits and Systems, vol. 38, pp. 389 1988. - 398, 1991.

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